(1)

\(\subseteq\)

\begin{align*} \left(\prod_{\lambda\in\Lambda}A_{\lambda}\right)\cup\left(\prod_{\mu\in\Lambda}B_{\mu}\right) & =\left\{ \left(a_{\lambda}\right)_{\lambda\in\Lambda};\lambda\in\Lambda,a_{\lambda}\in A_{\lambda}\right\} \cup\left\{ \left(b_{\mu}\right)_{\mu\in\Lambda};\mu\in\Lambda,b_{\mu}\in B_{\mu}\right\} \\ & =\left\{ \left(a_{\lambda}\right)_{\lambda\in\Lambda};\lambda\in\Lambda,a_{\lambda}\in A_{\lambda}\right\} \cup\left\{ \left(a_{\lambda}\right)_{\lambda\in\Lambda};\mu\in\Lambda,a_{\lambda}\in B_{\lambda}\right\} \\ & =\left\{ \left(a_{\lambda}\right)_{\lambda\in\Lambda};\left(\lambda\in\Lambda,a_{\lambda}\in A_{\lambda}\right)\lor\left(\lambda\in\Lambda,a_{\lambda}\in B_{\lambda}\right)\right\} \\ & \subseteq\left\{ \left(a_{\lambda}\right)_{\lambda\in\Lambda};\lambda\in\Lambda,a_{\lambda}\in A_{\lambda}\lor a_{\lambda}\in B_{\lambda}\right\} \cmt{\because\left(P_{1}\land P_{2}\right)\lor\left(Q_{1}\land Q_{2}\right)\Rightarrow\left(P_{1}\lor Q_{1}\right)\land\left(P_{2}\lor Q_{2}\right)}\\ & =\left\{ \left(a_{\lambda}\right)_{\lambda\in\Lambda};\lambda\in\Lambda,a_{\lambda}\in A_{\lambda}\cup B_{\lambda}\right\} \\ & =\prod_{\lambda\in\Lambda}\left(A_{\lambda}\cup B_{\lambda}\right) \end{align*} となるので\(\subseteq\)が成り立つ。

\(\supseteq\)は一般的に成り立たない

反例で示す。
\(\Lambda=\left\{ 1,2\right\} \)として、\(A_{1}=B_{2}=\emptyset\)とすると左辺は、
\begin{align*} \left(\prod_{\lambda\in\Lambda}A_{\lambda}\right)\cup\left(\prod_{\mu\in\Lambda}B_{\mu}\right) & =\left(A_{1}\times A_{2}\right)\cup\left(B_{1}\times B_{2}\right)\\ & =\left(\emptyset\times A_{2}\right)\cup\left(B_{1}\times\emptyset\right)\\ & =\emptyset\cup\emptyset\\ & =\emptyset \end{align*} となり、右辺は、
\begin{align*} \prod_{\lambda\in\Lambda}\left(A_{\lambda}\cup B_{\lambda}\right) & =\left(A_{1}\cup B_{1}\right)\times\left(A_{2}\cup B_{2}\right)\\ & =\left(\emptyset\cup B_{1}\right)\times\left(A_{2}\cup\emptyset\right)\\ & =B_{1}\times A_{2} \end{align*} となるので\(\subsetneq\)が成り立つ。
従って、\(\supseteq\)は一般的に成り立たない。

(2)

\begin{align*} \left(\prod_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\prod_{\mu\in\Lambda}B_{\mu}\right) & =\left\{ \left(x_{\lambda}\right)_{\lambda\in\Lambda};x_{\lambda}\in A_{\lambda}\right\} \cap\left\{ \left(x_{\lambda}\right)_{\lambda\in\Lambda};x_{\lambda}\in B_{\lambda}\right\} \\ & =\left\{ \left(x_{\lambda}\right)_{\lambda\in\Lambda};x_{\lambda}\in A_{\lambda}\land x_{\lambda}\in B_{\lambda}\right\} \\ & =\left\{ \left(x_{\lambda}\right)_{\lambda\in\Lambda};x_{\lambda}\in A_{\lambda}\cap B_{\lambda}\right\} \\ & =\prod_{\lambda\in\Lambda}\left(A_{\lambda}\cap B_{\lambda}\right) \end{align*}

(3)

\begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\times\left(\bigcup_{\mu\in M}B_{\mu}\right) & =\left\{ \left(a,b\right);\left(a\in\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\land\left(b\in\bigcup_{\mu\in M}B_{\mu}\right)\right\} \\ & =\left\{ \left(a,b\right);\left(\bigvee_{\lambda\in\Lambda}a\in A_{\lambda}\right)\land\left(\bigvee_{\mu\in M}b\in B_{\mu}\right)\right\} \\ & =\bigcup_{\lambda\in\Lambda}\bigcup_{\mu\in M}\left\{ \left(a,b\right);a\in A_{\lambda}\land b\in B_{\mu}\right\} \\ & =\bigcup_{\left(\lambda,\mu\right)\in\Lambda\times M}\left(A_{\lambda}\times B_{\mu}\right) \end{align*}

(4)

\begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\times\left(\bigcap_{\mu\in M}B_{\mu}\right) & =\left\{ \left(a,b\right);\left(a\in\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\land\left(b\in\bigcap_{\mu\in M}B_{\mu}\right)\right\} \\ & =\left\{ \left(a,b\right);\left(\bigwedge_{\lambda\in\Lambda}a\in A_{\lambda}\right)\land\left(\bigwedge_{\mu\in M}b\in B_{\mu}\right)\right\} \\ & =\bigcap_{\lambda\in\Lambda}\bigcap_{\mu\in M}\left\{ \left(a,b\right);a\in A_{\lambda}\land b\in B_{\mu}\right\} \\ & =\bigcap_{\left(\lambda,\mu\right)\in\Lambda\times M}\left(A_{\lambda}\times B_{\mu}\right) \end{align*}

(5)

\begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cup\left(\bigcup_{\mu\in M}B_{\mu}\right) & =\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cup\left(\bigcup_{\mu\in M}B_{\mu}\right)\right)\\ & =\bigcup_{\lambda\in\Lambda}\bigcup_{\mu\in M}\left(A_{\lambda}\cup B_{\mu}\right)\\ & =\bigcup_{\left(\lambda,\mu\right)\in\Lambda\times M}\left(A_{\lambda}\cup B_{\mu}\right) \end{align*}

(6)

\begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cup\left(\bigcap_{\mu\in M}B_{\mu}\right) & =\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cup\left(\bigcap_{\mu\in M}B_{\mu}\right)\right)\\ & =\bigcup_{\lambda\in\Lambda}\bigcap_{\mu\in M}\left(A_{\lambda}\cup B_{\mu}\right) \end{align*}

(7)

\begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cup\left(\bigcup_{\mu\in M}B_{\mu}\right) & =\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cup\left(\bigcup_{\mu\in M}B_{\mu}\right)\right)\\ & =\bigcap_{\lambda\in\Lambda}\bigcup_{\mu\in M}\left(A_{\lambda}\cup B_{\mu}\right) \end{align*}

(8)

\begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cup\left(\bigcap_{\mu\in M}B_{\mu}\right) & =\bigcap_{\lambda\in\Lambda}A_{\lambda}\cup\left(\bigcap_{\mu\in M}B_{\mu}\right)\\ & =\bigcap_{\lambda\in\Lambda}\bigcap_{\mu\in M}\left(A_{\lambda}\cup B_{\mu}\right)\\ & =\bigcap_{\left(\lambda,\mu\right)\in\Lambda\times M}\left(A_{\lambda}\cup B_{\mu}\right) \end{align*}

(8)-2

\begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cup\left(\bigcap_{\mu\in M}B_{\mu}\right) & =\left\{ a;\left(\bigwedge_{\lambda\in\Lambda}a\in A_{\lambda}\right)\lor\left(\bigwedge_{\mu\in M}a\in B_{\mu}\right)\right\} \\ & =\bigcap_{\lambda\in\Lambda}\bigcap_{\mu\in M}\left\{ a;a\in A_{\lambda}\lor a\in B_{\mu}\right\} \\ & =\bigcap_{\lambda\in\Lambda}\bigcap_{\mu\in M}\left\{ a;a\in\left(A_{\lambda}\cup B_{\mu}\right)\right\} \\ & =\bigcap_{\left(\lambda,\mu\right)\in\Lambda\times M}\left(A_{\lambda}\cup B_{\mu}\right) \end{align*}

(9)

\begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in M}B_{\mu}\right) & =\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cap\left(\bigcup_{\mu\in M}B_{\mu}\right)\right)\\ & =\bigcup_{\lambda\in\Lambda}\bigcup_{\mu\in M}\left(A_{\lambda}\cap B_{\mu}\right)\\ & =\bigcup_{\left(\lambda,\mu\right)\in\Lambda\times M}\left(A_{\lambda}\cap B_{\mu}\right) \end{align*}

(9)-2

\begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in M}B_{\mu}\right) & =\left\{ a;\left(\bigvee_{\lambda\in\Lambda}a\in A_{\lambda}\right)\land\left(\bigvee_{\mu\in M}a\in B_{\mu}\right)\right\} \\ & =\bigcup_{\lambda\in\Lambda}\bigcup_{\mu\in M}\left\{ a;a\in A_{\lambda}\land a\in B_{\mu}\right\} \\ & =\bigcup_{\lambda\in\Lambda}\bigcup_{\mu\in M}\left\{ a;a\in\left(A_{\lambda}\cap B_{\mu}\right)\right\} \\ & =\bigcup_{\left(\lambda,\mu\right)\in\Lambda\times M}\left(A_{\lambda}\cap B_{\mu}\right) \end{align*}

(10)

\begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcap_{\mu\in M}B_{\mu}\right) & =\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cap\left(\bigcap_{\mu\in M}B_{\mu}\right)\right)\\ & =\bigcup_{\lambda\in\Lambda}\bigcap_{\mu\in M}\left(A_{\lambda}\cap B_{\mu}\right) \end{align*}

(11)

\begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in M}B_{\mu}\right) & =\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cap\left(\bigcup_{\mu\in M}B_{\mu}\right)\right)\\ & =\bigcap_{\lambda\in\Lambda}\bigcup_{\mu\in M}\left(A_{\lambda}\cap B_{\mu}\right) \end{align*}

(12)

\begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcap_{\mu\in M}B_{\mu}\right) & =\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cap\left(\bigcap_{\mu\in M}B_{\mu}\right)\right)\\ & =\bigcap_{\lambda\in\Lambda}\bigcap_{\mu\in M}\left(A_{\lambda}\cap B_{\mu}\right)\\ & =\bigcap_{\left(\lambda,\mu\right)\in\Lambda\times M}\left(A_{\lambda}\cap B_{\mu}\right) \end{align*}

(13)

\begin{align*} \bigcup_{\lambda\in\Lambda}A_{\lambda}\cup\bigcup_{\lambda\in\Lambda}B_{\lambda} & =\left\{ x;x\in\bigcup_{\lambda\in\Lambda}A_{\lambda}\right\} \cup\left\{ x;x\in\bigcup_{\lambda\in\Lambda}B_{\lambda}\right\} \\ & =\left\{ x;x\in\bigcup_{\lambda\in\Lambda}A_{\lambda}\lor x\in\bigcup_{\lambda\in\Lambda}B_{\lambda}\right\} \\ & =\left\{ x;\left(\bigvee_{\lambda\in\Lambda}x\in A_{\lambda}\right)\lor\left(\bigvee_{\lambda\in\Lambda}x\in B_{\lambda}\right)\right\} \\ & =\left\{ x;\bigvee_{\lambda\in\Lambda}\left(x\in A_{\lambda}\lor x\in B_{\lambda}\right)\right\} \\ & =\left\{ x;\bigvee_{\lambda\in\Lambda}x\in\left(A_{\lambda}\cup B_{\lambda}\right)\right\} \\ & =\bigcup_{\lambda\in\Lambda}\left\{ x;x\in A_{\lambda}\cup B_{\lambda}\right\} \\ & =\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cup B_{\lambda}\right) \end{align*}

(14)

\begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cup\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right) & =\bigcap_{\mu\in\Lambda}\left(\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cup B_{\mu}\right)\\ & \supseteq\bigcap_{\mu\in\Lambda}\left(A_{\mu}\cup B_{\mu}\right)\cmt{\because A_{\mu}\subseteq\bigcup_{\lambda\in\Lambda}A_{\lambda}} \end{align*} \begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cup\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right) & =\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cup\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right)\right)\\ & \subseteq\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cup B_{\lambda}\right)\cmt{\because\bigcap_{\mu\in\Lambda}B_{\mu}\subseteq B_{\lambda}} \end{align*} これより、
\[ \bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cup B_{\lambda}\right)\subseteq\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cup\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right)\subseteq\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cup B_{\lambda}\right) \] となる。

(15)

\begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cup\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right) & =\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cup\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right)\right)\\ & \supseteq\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cup B_{\lambda}\right)\cmt{\because B_{\lambda}\subseteq\bigcup_{\mu\in\Lambda}B_{\mu}} \end{align*} \begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cup\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right) & =\bigcup_{\mu\in\Lambda}\left(\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cup B_{\mu}\right)\\ & \subseteq\bigcup_{\mu\in\Lambda}\left(A_{\mu}\cup B_{\mu}\right)\cmt{\because\bigcap_{\lambda\in\Lambda}A_{\lambda}\subseteq A_{\mu}} \end{align*} これより、
\[ \bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cup B_{\lambda}\right)\subseteq\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cup\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right)\subseteq\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cup B_{\lambda}\right) \] となる。

(16)

\begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cup\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right) & =\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cup\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right)\right)\\ & =\bigcap_{\lambda\in\Lambda}\bigcap_{\mu\in\Lambda}\left(A_{\lambda}\cup B_{\mu}\right)\\ & =\bigcap_{\left(\mu,\lambda\right)\in\Lambda\times\Lambda}\left(A_{\lambda}\cup B_{\mu}\right)\\ & =\bigcap_{\lambda=\mu}\left(A_{\lambda}\cup B_{\mu}\right)\cap\bigcap_{\lambda\ne\mu}\left(A_{\lambda}\cup B_{\mu}\right)\\ & \subseteq\bigcap_{\lambda=\mu}\left(A_{\lambda}\cup B_{\mu}\right)\\ & =\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cup B_{\lambda}\right) \end{align*}

(17)

\begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right) & =\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cap\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right)\right)\\ & =\bigcup_{\lambda\in\Lambda}\bigcup_{\mu\in\Lambda}\left(A_{\lambda}\cap B_{\mu}\right)\\ & =\bigcup_{\left(\mu,\lambda\right)\in\Lambda\times\Lambda}\left(A_{\lambda}\cap B_{\mu}\right)\\ & =\bigcup_{\lambda=\mu}\left(A_{\lambda}\cap B_{\mu}\right)\cup\bigcup_{\lambda\ne\mu}\left(A_{\lambda}\cap B_{\mu}\right)\\ & \supseteq\bigcup_{\lambda=\mu}\left(A_{\lambda}\cap B_{\mu}\right)\\ & =\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cap B_{\lambda}\right) \end{align*}

(18)

\begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right) & =\bigcap_{\mu\in\Lambda}\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap B_{\mu}\\ & \supseteq\bigcap_{\mu\in\Lambda}\left(A_{\mu}\cap B_{\mu}\right)\cmt{\because A_{\mu}\subseteq\bigcup_{\lambda\in\Lambda}A_{\lambda}} \end{align*} \begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right) & =\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cap\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right)\right)\\ & \subseteq\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cap B_{\lambda}\right)\cmt{\because\bigcap_{\mu\in\Lambda}B_{\mu}\subseteq B_{\lambda}} \end{align*} これより、
\[ \bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cap B_{\lambda}\right)\subseteq\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right)\subseteq\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cap B_{\lambda}\right) \] となる。

(19)

\begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right) & =\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cap\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right)\right)\\ & \supseteq\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cap B_{\lambda}\right)\cmt{\because B_{\lambda}\subseteq\bigcup_{\mu\in\Lambda}B_{\mu}} \end{align*} \begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right) & =\bigcup_{\mu\in\Lambda}\left(\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap B_{\mu}\right)\\ & \subseteq\bigcup_{\mu\in\Lambda}\left(A_{\mu}\cap B_{\mu}\right)\cmt{\because\bigcap_{\lambda\in\Lambda}A_{\lambda}\subseteq A_{\mu}} \end{align*} これより、
\[ \bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cap B_{\lambda}\right)\subseteq\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right)\subseteq\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cap B_{\lambda}\right) \] となる。

(20)

\begin{align*} \bigcap_{\lambda\in\Lambda}A_{\lambda}\cap\bigcap_{\lambda\in\Lambda}B_{\lambda} & =\left\{ x;x\in\bigcap_{\lambda\in\Lambda}A_{\lambda}\right\} \cap\left\{ x;x\in\bigcap_{\lambda\in\Lambda}B_{\lambda}\right\} \\ & =\left\{ x;x\in\bigcap_{\lambda\in\Lambda}A_{\lambda}\land x\in\bigcap_{\lambda\in\Lambda}B_{\lambda}\right\} \\ & =\left\{ x;\left(\bigwedge_{\lambda\in\Lambda}x\in A_{\lambda}\right)\land\left(\bigwedge_{\lambda\in\Lambda}x\in B_{\lambda}\right)\right\} \\ & =\left\{ x;\bigwedge_{\lambda\in\Lambda}\left(x\in A_{\lambda}\land x\in B_{\lambda}\right)\right\} \\ & =\left\{ x;\bigwedge_{\lambda\in\Lambda}x\in\left(A_{\lambda}\cap B_{\lambda}\right)\right\} \\ & =\bigcap_{\lambda\in\Lambda}\left\{ x;x\in A_{\lambda}\cap B_{\lambda}\right\} \\ & =\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cap B_{\lambda}\right) \end{align*}

(21)

\begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\setminus\left(\bigcup_{\mu\in M}B_{\mu}\right) & =\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in M}B_{\mu}\right)^{c}\\ & =\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcap_{\mu\in M}B_{\mu}^{c}\right)\\ & =\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cap\left(\bigcap_{\mu\in M}B_{\mu}^{c}\right)\right)\\ & =\bigcup_{\lambda\in\Lambda}\bigcap_{\mu\in M}\left(A_{\lambda}\cap B_{\mu}^{c}\right)\\ & =\bigcup_{\lambda\in\Lambda}\bigcap_{\mu\in M}\left(A_{\lambda}\setminus B_{\mu}\right) \end{align*}

(22)

\begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\setminus\left(\bigcap_{\mu\in M}B_{\mu}\right) & =\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcap_{\mu\in M}B_{\mu}\right)^{c}\\ & =\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in M}B_{\mu}^{c}\right)\\ & =\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cap\left(\bigcup_{\mu\in M}B_{\mu}^{c}\right)\right)\\ & =\bigcup_{\lambda\in\Lambda}\bigcup_{\mu\in M}\left(A_{\lambda}\cap B_{\mu}^{c}\right)\\ & =\bigcup_{\left(\lambda,\mu\right)\in\Lambda\times M}\left(A_{\lambda}\setminus B_{\mu}\right) \end{align*}

(23)

\begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\setminus\left(\bigcup_{\mu\in M}B_{\mu}\right) & =\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in M}B_{\mu}\right)^{c}\\ & =\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcap_{\mu\in M}B_{\mu}^{c}\right)\\ & =\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cap\left(\bigcap_{\mu\in M}B_{\mu}^{c}\right)\right)\\ & =\bigcap_{\lambda\in\Lambda}\bigcap_{\mu\in M}\left(A_{\lambda}\cap B_{\mu}^{c}\right)\\ & =\bigcap_{\left(\lambda,\mu\right)\in\Lambda\times M}\left(A_{\lambda}\setminus B_{\mu}\right) \end{align*}

(24)

\begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\setminus\left(\bigcap_{\mu\in M}B_{\mu}\right) & =\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcap_{\mu\in M}B_{\mu}\right)^{c}\\ & =\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in M}B_{\mu}^{c}\right)\\ & =\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cap\left(\bigcup_{\mu\in M}B_{\mu}^{c}\right)\right)\\ & =\bigcap_{\lambda\in\Lambda}\bigcup_{\mu\in M}\left(A_{\lambda}\cap B_{\mu}^{c}\right)\\ & =\bigcap_{\lambda\in\Lambda}\bigcup_{\mu\in M}\left(A_{\lambda}\setminus B_{\mu}\right) \end{align*}

(25)

\begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\setminus\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right) & =\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right)^{c}\\ & =\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcap_{\mu\in\Lambda}B_{\mu}^{c}\right)\\ & =\bigcap_{\mu\in\Lambda}\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap B_{\mu}^{c}\\ & \supseteq\bigcap_{\mu\in\Lambda}\left(A_{\mu}\cap B_{\mu}^{c}\right)\cmt{\because A_{\mu}\subseteq\bigcup_{\lambda\in\Lambda}A_{\lambda}}\\ & =\bigcap_{\mu\in\Lambda}\left(A_{\mu}\setminus B_{\mu}\right) \end{align*} \begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\setminus\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right) & =\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right)^{c}\\ & =\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcap_{\mu\in\Lambda}B_{\mu}^{c}\right)\\ & =\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cap\left(\bigcap_{\mu\in\Lambda}B_{\mu}^{c}\right)\right)\\ & \subseteq\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cap B_{\lambda}^{c}\right)\cmt{\because\bigcap_{\mu\in\Lambda}B_{\mu}^{c}\subseteq B_{\lambda}^{c}}\\ & =\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\setminus B_{\lambda}\right) \end{align*} これより、
\[ \bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\setminus B_{\lambda}\right)\subseteq\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\setminus\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right)\subseteq\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\setminus B_{\lambda}\right) \] となる。

(26)

\begin{align*} \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\setminus\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right) & =\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right)^{c}\\ & =\left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in\Lambda}B_{\mu}^{c}\right)\\ & =\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cap\left(\bigcup_{\mu\in\Lambda}B_{\mu}^{c}\right)\right)\\ & \supseteq\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\cap B_{\lambda}^{c}\right)\cmt{\because B_{\lambda}^{c}\subseteq\bigcup_{\mu\in\Lambda}B_{\mu}^{c}}\\ & =\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\setminus B_{\lambda}\right) \end{align*} これより、
\[ \left(\bigcup_{\lambda\in\Lambda}A_{\lambda}\right)\setminus\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right)\supseteq\bigcup_{\lambda\in\Lambda}\left(A_{\lambda}\setminus B_{\lambda}\right) \] となる。

(27)

\begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\setminus\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right) & =\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in\Lambda}B_{\mu}\right)^{c}\\ & =\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcap_{\mu\in\Lambda}B_{\mu}^{c}\right)\\ & =\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cap B_{\lambda}^{c}\right)\\ & =\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\setminus B_{\lambda}\right) \end{align*}

(28)

\begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\setminus\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right) & =\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right)^{c}\\ & =\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in\Lambda}B_{\mu}^{c}\right)\\ & =\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cap\left(\bigcup_{\mu\in\Lambda}B_{\mu}^{c}\right)\right)\\ & \supseteq\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\cap B_{\lambda}^{c}\right)\cmt{\because B_{\lambda}^{c}\subseteq\bigcup_{\mu\in\Lambda}B_{\mu}^{c}}\\ & =\bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\setminus B_{\lambda}\right) \end{align*} \begin{align*} \left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\setminus\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right) & =\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right)^{c}\\ & =\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap\left(\bigcup_{\mu\in\Lambda}B_{\mu}^{c}\right)\\ & =\bigcup_{\mu\in\Lambda}\left(\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\cap B_{\mu}^{c}\right)\\ & \subseteq\bigcup_{\mu\in\Lambda}\left(A_{\mu}\cap B_{\mu}^{c}\right)\cmt{\because\bigcap_{\lambda\in\Lambda}A_{\lambda}\subseteq A_{\mu}}\\ & =\bigcup_{\mu\in\Lambda}\left(A_{\mu}\setminus B_{\mu}\right) \end{align*} となるので、
\[ \bigcap_{\lambda\in\Lambda}\left(A_{\lambda}\setminus B_{\lambda}\right)\subseteq\left(\bigcap_{\lambda\in\Lambda}A_{\lambda}\right)\setminus\left(\bigcap_{\mu\in\Lambda}B_{\mu}\right)\subseteq\bigcup_{\mu\in\Lambda}\left(A_{\mu}\setminus B_{\mu}\right) \] となる。